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Theorem lsmelvalm 14978
Description: Subgroup sum membership analog of lsmelval 14976 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmelvalm.m  |-  .-  =  ( -g `  G )
lsmelvalm.p  |-  .(+)  =  (
LSSum `  G )
lsmelvalm.t  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
lsmelvalm.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
Assertion
Ref Expression
lsmelvalm  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .-  z ) ) )
Distinct variable groups:    y, z,  .-    y, G, z    ph, y,
z    y, T, z    y, U, z    y, X, z
Allowed substitution hints:    .(+) ( y, z)

Proof of Theorem lsmelvalm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lsmelvalm.t . . 3  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
2 lsmelvalm.u . . 3  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
3 eqid 2296 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
4 lsmelvalm.p . . . 4  |-  .(+)  =  (
LSSum `  G )
53, 4lsmelval 14976 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. x  e.  U  X  =  ( y
( +g  `  G ) x ) ) )
61, 2, 5syl2anc 642 . 2  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. x  e.  U  X  =  ( y ( +g  `  G ) x ) ) )
72adantr 451 . . . . . . . 8  |-  ( (
ph  /\  y  e.  T )  ->  U  e.  (SubGrp `  G )
)
8 eqid 2296 . . . . . . . . 9  |-  ( inv g `  G )  =  ( inv g `  G )
98subginvcl 14646 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  G )  /\  x  e.  U )  ->  (
( inv g `  G ) `  x
)  e.  U )
107, 9sylan 457 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
( inv g `  G ) `  x
)  e.  U )
11 eqid 2296 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
12 lsmelvalm.m . . . . . . . . 9  |-  .-  =  ( -g `  G )
13 subgrcl 14642 . . . . . . . . . . 11  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
141, 13syl 15 . . . . . . . . . 10  |-  ( ph  ->  G  e.  Grp )
1514ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  G  e.  Grp )
1611subgss 14638 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
171, 16syl 15 . . . . . . . . . . 11  |-  ( ph  ->  T  C_  ( Base `  G ) )
1817sselda 3193 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  T )  ->  y  e.  ( Base `  G
) )
1918adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  y  e.  ( Base `  G
) )
2011subgss 14638 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
217, 20syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  T )  ->  U  C_  ( Base `  G
) )
2221sselda 3193 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  x  e.  ( Base `  G
) )
2311, 3, 12, 8, 15, 19, 22grpsubinv 14557 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
y  .-  ( ( inv g `  G ) `
 x ) )  =  ( y ( +g  `  G ) x ) )
2423eqcomd 2301 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
y ( +g  `  G
) x )  =  ( y  .-  (
( inv g `  G ) `  x
) ) )
25 oveq2 5882 . . . . . . . . 9  |-  ( z  =  ( ( inv g `  G ) `
 x )  -> 
( y  .-  z
)  =  ( y 
.-  ( ( inv g `  G ) `
 x ) ) )
2625eqeq2d 2307 . . . . . . . 8  |-  ( z  =  ( ( inv g `  G ) `
 x )  -> 
( ( y ( +g  `  G ) x )  =  ( y  .-  z )  <-> 
( y ( +g  `  G ) x )  =  ( y  .-  ( ( inv g `  G ) `  x
) ) ) )
2726rspcev 2897 . . . . . . 7  |-  ( ( ( ( inv g `  G ) `  x
)  e.  U  /\  ( y ( +g  `  G ) x )  =  ( y  .-  ( ( inv g `  G ) `  x
) ) )  ->  E. z  e.  U  ( y ( +g  `  G ) x )  =  ( y  .-  z ) )
2810, 24, 27syl2anc 642 . . . . . 6  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  E. z  e.  U  ( y
( +g  `  G ) x )  =  ( y  .-  z ) )
29 eqeq1 2302 . . . . . . 7  |-  ( X  =  ( y ( +g  `  G ) x )  ->  ( X  =  ( y  .-  z )  <->  ( y
( +g  `  G ) x )  =  ( y  .-  z ) ) )
3029rexbidv 2577 . . . . . 6  |-  ( X  =  ( y ( +g  `  G ) x )  ->  ( E. z  e.  U  X  =  ( y  .-  z )  <->  E. z  e.  U  ( y
( +g  `  G ) x )  =  ( y  .-  z ) ) )
3128, 30syl5ibrcom 213 . . . . 5  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  ( X  =  ( y
( +g  `  G ) x )  ->  E. z  e.  U  X  =  ( y  .-  z
) ) )
3231rexlimdva 2680 . . . 4  |-  ( (
ph  /\  y  e.  T )  ->  ( E. x  e.  U  X  =  ( y
( +g  `  G ) x )  ->  E. z  e.  U  X  =  ( y  .-  z
) ) )
338subginvcl 14646 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  G )  /\  z  e.  U )  ->  (
( inv g `  G ) `  z
)  e.  U )
347, 33sylan 457 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  (
( inv g `  G ) `  z
)  e.  U )
3518adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  y  e.  ( Base `  G
) )
3621sselda 3193 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  z  e.  ( Base `  G
) )
3711, 3, 8, 12grpsubval 14541 . . . . . . . 8  |-  ( ( y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
y  .-  z )  =  ( y ( +g  `  G ) ( ( inv g `  G ) `  z
) ) )
3835, 36, 37syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  (
y  .-  z )  =  ( y ( +g  `  G ) ( ( inv g `  G ) `  z
) ) )
39 oveq2 5882 . . . . . . . . 9  |-  ( x  =  ( ( inv g `  G ) `
 z )  -> 
( y ( +g  `  G ) x )  =  ( y ( +g  `  G ) ( ( inv g `  G ) `  z
) ) )
4039eqeq2d 2307 . . . . . . . 8  |-  ( x  =  ( ( inv g `  G ) `
 z )  -> 
( ( y  .-  z )  =  ( y ( +g  `  G
) x )  <->  ( y  .-  z )  =  ( y ( +g  `  G
) ( ( inv g `  G ) `
 z ) ) ) )
4140rspcev 2897 . . . . . . 7  |-  ( ( ( ( inv g `  G ) `  z
)  e.  U  /\  ( y  .-  z
)  =  ( y ( +g  `  G
) ( ( inv g `  G ) `
 z ) ) )  ->  E. x  e.  U  ( y  .-  z )  =  ( y ( +g  `  G
) x ) )
4234, 38, 41syl2anc 642 . . . . . 6  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  E. x  e.  U  ( y  .-  z )  =  ( y ( +g  `  G
) x ) )
43 eqeq1 2302 . . . . . . 7  |-  ( X  =  ( y  .-  z )  ->  ( X  =  ( y
( +g  `  G ) x )  <->  ( y  .-  z )  =  ( y ( +g  `  G
) x ) ) )
4443rexbidv 2577 . . . . . 6  |-  ( X  =  ( y  .-  z )  ->  ( E. x  e.  U  X  =  ( y
( +g  `  G ) x )  <->  E. x  e.  U  ( y  .-  z )  =  ( y ( +g  `  G
) x ) ) )
4542, 44syl5ibrcom 213 . . . . 5  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  ( X  =  ( y  .-  z )  ->  E. x  e.  U  X  =  ( y ( +g  `  G ) x ) ) )
4645rexlimdva 2680 . . . 4  |-  ( (
ph  /\  y  e.  T )  ->  ( E. z  e.  U  X  =  ( y  .-  z )  ->  E. x  e.  U  X  =  ( y ( +g  `  G ) x ) ) )
4732, 46impbid 183 . . 3  |-  ( (
ph  /\  y  e.  T )  ->  ( E. x  e.  U  X  =  ( y
( +g  `  G ) x )  <->  E. z  e.  U  X  =  ( y  .-  z
) ) )
4847rexbidva 2573 . 2  |-  ( ph  ->  ( E. y  e.  T  E. x  e.  U  X  =  ( y ( +g  `  G
) x )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .-  z
) ) )
496, 48bitrd 244 1  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .-  z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   inv gcminusg 14379   -gcsg 14381  SubGrpcsubg 14631   LSSumclsm 14961
This theorem is referenced by:  lsmelvalmi  14979  pgpfac1lem2  15326  pgpfac1lem3  15328  pgpfac1lem4  15329  mapdpglem3  32487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-lsm 14963
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